In this subsection we record some rather straightforward applications of the
previous results for
collections of Hardy field sequences (defined in Section REF ) that satisfy some
growth assumptions that we define next. Our method also gives alternative proofs of known results from [1]}, [2]}, [3]}, [4]} that avoid the theory of characteristic factors and equidistribution results on nilmanifolds. For the purposes of this sub-section we assume that all Hardy fields \(\mathcal {H}\)  considered have
the following property:
\(\text{if } a,b\in \mathcal {H }, \text{ then } a\circ b^{-1}\in \mathcal {H } \text{ and } a(\cdot +h)\in \mathcal {H} \text{ for every } h\in {\mathbb {R}}_+.\) 
